Theorem nmlnopne0

Description: A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	nmlnopne0	⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) ≠ 0 ↔ 𝑇 ≠ 0hop ) )

Proof

Step	Hyp	Ref	Expression
1		nmlnop0	⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) = 0 ↔ 𝑇 = 0hop ) )
2	1	necon3bid	⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) ≠ 0 ↔ 𝑇 ≠ 0hop ) )